# Definition:Finer Filter on Set

(Redirected from Definition:Superfilter)

## Definition

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\mathcal F, \mathcal F' \subset \powerset S$ be two filters on $S$.

Let $\mathcal F \subseteq \mathcal F'$.

Then $\mathcal F'$ is finer than $\mathcal F$.

### Strictly Finer

Let $\mathcal F \subset \mathcal F'$, that is, $\mathcal F \subseteq \mathcal F'$ but $\mathcal F \ne \mathcal F'$.

Then $\mathcal F'$ is strictly finer than $\mathcal F$.

## Also known as

A finer filter than $\mathcal F$ can also be referred to as a superfilter of $\mathcal F$.